Planning a Neural Network
First of all what you are trying to do can be achieved by simple logical programming. Secondly you are making things overly complex.

1 node in a Neural Net can predict as many values as you would like it to, as it outputs a real number no matter the activation function. It is us who round of the number to 0 or 1 or maybe 0, 0.5, 1 depending on the classification job at hand.

2 Neurons can predict 4 classes 00, 01, 10, 11 depending on how you use it. Now in your specific of quadrant detection 2 Neurons are sufficient, since not only can you decompose the problem into two separate problems, both the problems are linearly separable too. This is not always the case or it maybe too complex to find linearly separable classes, so in general one output node is reserved for 1 class only. If it is that class only that output node is activated while the rest is 0. Memory and process consuming, but easier.

Coming to your first approach. I don't exactly how hope to achieve 4 class classification with a single output node. Minimum of 2 nodes is required. The problem with your approach is that if you think critically your hidden nodes are not doing anything other than scaling the inputs, so almost the same type input is propagated and then you are asking a node to perform a a 4 task classification which is not linearly separable as per me. sin will not be able to separate it if you think critically.

The second idea is better, just remove the hidden node and you will be able to classify it based on 00, 01, 10, 11 outputs. Although this problem can be solved by simple logical computation and your Neural Net is basically doing nothing.

Please point out to me if you find any mathematical mistakes. Hope this helps!